From 548f860cd45c5950a681f8e264e216d5284d3033 Mon Sep 17 00:00:00 2001
From: "Brady J. Garvin" <bgarvin@cse.unl.edu>
Date: Tue, 1 Nov 2022 12:00:12 -0500
Subject: [PATCH] Designed and implemented mergesort and quickmedian.
---
divide-and-conquer/notes.md | 73 +++++++++++--------
.../src/features/median/kthSmallest.js | 12 ++-
.../src/features/sorting/sorting.js | 22 +++++-
3 files changed, 75 insertions(+), 32 deletions(-)
diff --git a/divide-and-conquer/notes.md b/divide-and-conquer/notes.md
index cdbfed9..45ab4df 100644
--- a/divide-and-conquer/notes.md
+++ b/divide-and-conquer/notes.md
@@ -48,19 +48,19 @@ Problem: Sort a list in Θ(n log n) time.
```
In: [8, 7, 6, 9, 5]
- Out: […, …, …, …, …]
+ Out: [5, 6, 7, 8, 9]
/ \
/ \
- In: […] In: […]
- Out: […] Out: […]
+ In: [8, 7] In: [6, 9, 5]
+ Out: [7, 8] Out: [5, 6, 9]
/ \ / \
/ \ / \
-In: […] In: […] In: […] In: […]
-Out: […] Out: […] Out: […] Out: […]
+In: [8] In: [7] In: [6] In: [9, 5]
+Out: [8] Out: [7] Out: [6] Out: [5, 9]
/ \
/ \
- In: […] In: […]
- Out: […] Out: […]
+ In: [9] In: [5]
+ Out: [9] Out: [5]
```
## Analysis
@@ -69,14 +69,14 @@ Out: […] Out: […] Out: […] Out: […]
T(n) = T(⌊n/2⌋) + T(⌈n/2⌉) + Θ(n) for n ≫ 0
T(n) ≈ 2T(n/2) + Θ(n) for n ≫ 0
- T(n) = Θ(⋯)
+ T(n) = Θ(n log n)
* Uneven split:
T(n) = T(n - 1) + T(1) + Θ(n) for n ≫ 0
T(n) = T(n - 1) + Θ(1) + Θ(n) for n ≫ 0
T(n) = T(n - 1) + Θ(n) for n ≫ 0
- T(n) = Θ(⋯)
+ T(n) = Θ(n²)
--------------------------------------------------------------------------------
@@ -96,11 +96,11 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
```
In: [8, 7, 6, 9, 5]
- Out: …
+ Out: ????
/ \
/ \
- In: […] In: […]
- Out: … Out: …
+ In: [8, 7] In: [6, 9, 5]
+ Out: 7.5 Out: 6
```
* Option A: We are missing information from the recursive calls. Add output(s) so that they can return that information.
@@ -108,15 +108,15 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
## Design (second attempt)
-* More generally, we want …, so we add an input `k`:
+* More generally, we want kth smallest element, so we add an input `k`:
```
In: [8, 7, 6, 9, 5], k = 2
Out: …
/ \
/ \
- In: […], k = 2 In: […], k = 0
- Out: … Out: …
+ In: [8, 7], k = 2 In: [6, 9, 5], k = 0
+ Out: ???? Out: 5
```
## Partitioning
@@ -131,24 +131,39 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
```
In: [8, 7, 6, 9, 5], k = 2
- In: […], k = 2
- In: […], k = 2
- Split: […], …, […]
- Out: …
+ *
+ i
+ j
+ In: [8, 7, 6, 5, 9], k = 2
+ *
+ i (crossed)
+ j
+ i
+ In: [5, 7, 6, 8, 9], k = 2
+ Split: [5, 7, 6], 8, [9]
+ Out: 7
|
|
|
- In: […], k = …
- In: […], k = …
- Split: […], …, […]
- Out: …
+ In: [5, 7, 6], k = 2
+ *
+ i (crossed)
+ j
+ i
+ In: [5, 7, 6], k = 2
+ Split: [], 5, [7, 6]
+ Out: 7
|
|
|
- In: […], k = …
- In: […], k = …
- Split: […], …, […]
- Out: …
+ In: [7, 6], k = 1
+ *
+ i (crossed)
+ j
+ i
+ In: [6, 7], k = 1
+ Split: [6], 7, []
+ Out: 7
```
## Analysis
@@ -156,9 +171,9 @@ Problem: Find the median of an odd-length list in average-case Θ(n) time. Use
* Uneven split:
T(n) = T(n - 1) + Θ(n) for n ≫ 0
- T(n) = Θ(⋯)
+ T(n) = Θ(n²)
* Even split:
T(n) ≈ T(n/2) + Θ(n) for n ≫ 0
- T(n) = Θ(⋯)
+ T(n) = Θ(n)
diff --git a/divide-and-conquer/src/features/median/kthSmallest.js b/divide-and-conquer/src/features/median/kthSmallest.js
index 7125d1f..26e87bb 100644
--- a/divide-and-conquer/src/features/median/kthSmallest.js
+++ b/divide-and-conquer/src/features/median/kthSmallest.js
@@ -31,5 +31,15 @@ export function kthSmallest(sequence, k) {
k >= 0 && k < sequence.length,
'Tried to find the kth smallest element when k is out of bounds.',
);
- return sequence[0]; // TODO: stub
+ if (sequence.length < 2) {
+ return sequence[0];
+ }
+ const [left, pivot, right] = partition(sequence);
+ if (k === left.length) {
+ return pivot;
+ }
+ if (k < left.length) {
+ return kthSmallest(left, k);
+ }
+ return kthSmallest(right, k - left.length - 1);
}
diff --git a/divide-and-conquer/src/features/sorting/sorting.js b/divide-and-conquer/src/features/sorting/sorting.js
index bca3d88..10eb765 100644
--- a/divide-and-conquer/src/features/sorting/sorting.js
+++ b/divide-and-conquer/src/features/sorting/sorting.js
@@ -1,7 +1,25 @@
function merge(left, right) {
- return []; // TODO: stub
+ const results = [];
+ let i = 0;
+ let j = 0;
+ while (i < left.length || j < right.length) {
+ if (j >= right.length || (i < left.length && left[i] < right[j])) {
+ results.push(left[i]);
+ ++i;
+ } else {
+ results.push(right[j]);
+ ++j;
+ }
+ }
+ return results;
}
export function sort(list) {
- return list; // TODO: stub
+ if (list.length < 2) {
+ return list;
+ }
+ const middleIndex = Math.floor(list.length / 2);
+ const left = sort(list.slice(0, middleIndex));
+ const right = sort(list.slice(middleIndex, list.length));
+ return merge(left, right);
}
--
GitLab